Gear transmission stands as a fundamental method for power transmission, and researchers have conducted extensive studies in this field. The Variable Hyperbolic Circular-Arc-Tooth-Trace (VH-CATT) cylindrical gear is a novel type of gear developed based on the Gleason spiral bevel gear. This type of cylindrical gear exhibits high load-carrying capacity, high transmission efficiency, low installation precision requirements, and the absence of axial force. Current research on VH-CATT cylindrical gears encompasses areas such as forming principles, manufacturing processes, tooth contact analysis, meshing stiffness and impact forces, wear and lubrication, and tooth surface errors. However, research focusing on the tooth surface modification design and the analysis of dynamic meshing characteristics for VH-CATT cylindrical gears remains relatively limited, which considerably restricts their development and industrial application.
Modification design for gears can improve the load distribution on the meshing tooth flanks, enhance the load-bearing capacity of the gear pair, reduce transmission error, and ultimately achieve the goal of vibration and noise reduction. Based on the forming principle of VH-CATT cylindrical gears machined with a large cutter head, this paper proposes a tooth surface modification method involving tilting the cutter head during the machining process. The dynamic meshing characteristics of VH-CATT cylindrical gears under different modification parameters are subsequently analyzed. Firstly, the modified tooth surface equation is derived according to the forming principle, and the tooth surface is reconstructed. Secondly, kinematics analysis models for VH-CATT cylindrical gears with different modification parameters are established. Finally, kinematics and dynamics analyses are performed on the modified VH-CATT cylindrical gear pairs. The fluctuation of the driven wheel’s rotational speed and the evolution of dynamic meshing forces are calculated and examined. The results provide a theoretical foundation for vibration reduction, noise control, and modification design of these advanced cylindrical gears.
Mathematical Model and 3D Modeling of Modified VH-CATT Cylindrical Gears
Tooth Surface Mathematical Model
The VH-CATT cylindrical gear is manufactured using a dual-blade milling process with a large diameter cutter head. To enhance the dynamic meshing performance and reduce vibration and noise, a modification method involving tilting the cutter head during the milling process is proposed in this work. The coordinate system for the machining process with the tilted large cutter head is established.
Based on the gear meshing theory and the established coordinate systems, the mathematical model for the tooth surface of the modified VH-CATT cylindrical gear is derived. The position vector of a point on the generated tooth surface can be expressed as follows:
$$ \mathbf{r}_s(u, \theta, \phi_1) = \mathbf{M}_{s1}(\phi_1) \cdot \mathbf{M}_{1c} \cdot \mathbf{r}_c(u, \theta) $$
Where $\mathbf{r}_c(u, \theta)$ is the position vector of a point on the cutter surface in the cutter coordinate system $S_c$, $u$ and $\theta$ are the surface parameters of the cutter. $\mathbf{M}_{1c}$ is the transformation matrix from the cutter coordinate system $S_c$ to the gear blank coordinate system $S_1$, which incorporates the installation parameters including the nominal tooth trace radius $R_T$, the cutter tilt angle $\delta$, and the offset. $\mathbf{M}_{s1}(\phi_1)$ is the transformation matrix from $S_1$ to the fixed gear coordinate system $S_s$, involving the rotation angle $\phi_1$ of the gear blank. The specific form of the surface equation, considering the tilt, is:
$$
\mathbf{r}_s^{(i)}(u, \theta, \phi_1) =
\begin{bmatrix}
(R_T \pm u \sin \alpha_n) \cos(\theta \mp \frac{m \theta}{2R_T}) \cos(\phi_1 + \theta) \mp u \cos \alpha_n \sin(\phi_1 + \theta) + \Delta x \cos \phi_1 – \Delta y \sin \phi_1 \\
-(R_T \pm u \sin \alpha_n) \cos(\theta \mp \frac{m \theta}{2R_T}) \sin(\phi_1 + \theta) \mp u \cos \alpha_n \cos(\phi_1 + \theta) – \Delta x \sin \phi_1 – \Delta y \cos \phi_1 \\
(R_T \pm u \sin \alpha_n) \sin(\theta \mp \frac{m \theta}{2R_T}) + \Delta z \\
1
\end{bmatrix}
$$
In this equation, the upper sign corresponds to the convex side (generated by the inner blade of the cutter), and the lower sign corresponds to the concave side (generated by the outer blade). $\alpha_n$ is the normal pressure angle of the cutter, $m$ is the module, $R_T$ is the nominal radius of the circular-arc tooth trace, $u$ is the distance parameter on the cutter surface, $\theta$ is the rotational parameter of the cutter, $\delta$ is the tilt angle of the cutter head, and $\phi_1$ is the rotational angle of the gear blank. The terms $\Delta x$, $\Delta y$, and $\Delta z$ represent the setting parameters of the cutter head center, which are functions of $\delta$ and $R_T$.
The unit normal vector $\mathbf{n}_s$ on the tooth surface is essential for contact analysis and is given by:
$$ \mathbf{n}_s(u, \theta, \phi_1) = \frac{\partial \mathbf{r}_s}{\partial u} \times \frac{\partial \mathbf{r}_s}{\partial \theta} \bigg/ \left\| \frac{\partial \mathbf{r}_s}{\partial u} \times \frac{\partial \mathbf{r}_s}{\partial \theta} \right\| $$
Three-Dimensional Modeling
To perform dynamic analysis, precise three-dimensional models of the gear pair are required. The basic parameters for the VH-CATT cylindrical gear pair are selected as follows: number of teeth for the pinion $z_1=29$ and for the gear $z_2=41$, module $m=8 \text{ mm}$, normal pressure angle $\alpha_n=20^\circ$, face width $b=60 \text{ mm}$, and nominal cutter radius $R_c=180 \text{ mm}$. Using the derived mathematical model, numerical calculations are performed in MATLAB to obtain point cloud data for both the concave and convex flanks of a single tooth of the VH-CATT cylindrical gear. This data is then imported into 3D modeling software (e.g., UG NX) to generate the accurate solid models of the driving and driven gears, finally assembling them into the gear pair model.
Dynamic Simulation Modeling in Adams
To investigate the dynamic meshing characteristics of VH-CATT cylindrical gears with different modification parameters, multi-body dynamics simulations are conducted using Adams software. A virtual prototype model of the modified VH-CATT cylindrical gear pair is established within the Adams environment.
First, the 3D solid models of the gears, created previously, are imported. The material properties for the gear bodies are defined, typically using steel with a density of $\rho = 7850 \text{ kg/m}^3$, a Young’s modulus of $E = 2.07 \times 10^{11} \text{ Pa}$, and a Poisson’s ratio of $\nu = 0.3$. The driving and driven shafts are modeled, and the gear bodies are connected to their respective shafts using fixed joints. Revolute joints are applied between the shafts and the ground to allow rotational motion. A constant rotational speed (e.g., $1000 \text{ rpm}$) is applied to the driving shaft as the input motion. A constant load torque (e.g., $1500 \text{ N·m}$) is applied to the driven shaft to simulate the resisting load.
The most critical part of the modeling is defining the contact force between the meshing tooth flanks of the two cylindrical gears. In Adams, a solid contact force model is used to simulate the realistic meshing impact. The contact force is typically modeled using an impact function based on the Hertzian contact theory and a damping component. The normal contact force $F_n$ is calculated as:
$$ F_n = k \cdot g^e + c \cdot \dot{g} $$
Where $g$ is the penetration depth between the two contacting geometries, $\dot{g}$ is the penetration velocity, $k$ is the contact stiffness, $e$ is the force exponent (usually 1.5 for metal contact), and $c$ is the damping coefficient.
The contact stiffness $k$ is approximated based on the material properties and the geometry of the contacting bodies at the point of contact. For two cylindrical bodies in contact, the equivalent radius $R^*$ and equivalent modulus $E^*$ are calculated. The stiffness can be derived from the Hertzian contact theory. The formula for the stiffness coefficient used in the simulation is based on the following relationships:
$$ \frac{1}{R^*} = \frac{1}{R_1} \pm \frac{1}{R_2} $$
$$ \frac{1}{E^*} = \frac{1-\nu_1^2}{E_1} + \frac{1-\nu_2^2}{E_2} $$
For the simulated steel cylindrical gears, with $E_1 = E_2 = 2.07 \times 10^{11} \text{ Pa}$ and $\nu_1 = \nu_2 = 0.3$, the equivalent modulus $E^*$ is approximately $1.138 \times 10^{11} \text{ Pa}$. The specific stiffness value used in the Adams model is then determined based on this equivalent modulus and the estimated contact geometry of the curved tooth surfaces.
Influence of Modification Parameter on Speed Fluctuation
The dynamic transmission stability of a gear pair can be assessed by examining the fluctuation in the rotational speed of the driven wheel. A perfectly stable and rigid transmission would result in a constant output speed. However, due to factors like time-varying mesh stiffness and contact impacts, the output speed exhibits periodic fluctuations. The modification of the tooth surface aims to minimize these fluctuations.
To study the effect of the cutter tilt angle $\delta$ on the transmission stability of the VH-CATT cylindrical gear, a single-variable principle is adopted. Gear pair models are created with different tilt angles $\delta$ (0°, 3°, 5°, and 7°) while keeping all other geometric and simulation parameters constant. Each model is subjected to a dynamics simulation in Adams with a load torque of $1500 \text{ N·m}$ applied to the driven gear. The rotational speed of the driven gear is recorded over time.
The results show that the driven gear’s speed exhibits periodic oscillations for all modification cases, indicating inherent dynamic behavior in the cylindrical gear transmission. The key metrics—maximum speed, minimum speed, average speed, and speed deviation—are extracted from the simulation data. The speed deviation is calculated as a percentage of the average speed to quantify fluctuation magnitude.
| Cutter Tilt Angle $\delta$ (°) | Max Speed (deg/s) | Min Speed (deg/s) | Avg Speed (deg/s) | Speed Deviation (%) |
|---|---|---|---|---|
| 0 | 2657.71 | 2224.44 | 2476.28 | 0.027 |
| 3 | 2654.44 | 2225.73 | 2475.19 | 0.017 |
| 5 | 2656.56 | 2229.99 | 2475.89 | -0.011 |
| 7 | 2658.51 | 2241.21 | 2475.78 | -0.007 |
The table reveals that the average output speed remains nearly constant across different modification angles, with only minor variations. This confirms that the modification does not significantly alter the fundamental speed ratio of the cylindrical gear pair. However, the speed deviation, which reflects the amplitude of speed fluctuation, shows a clear trend. As the cutter tilt angle $\delta$ increases from 0° to 5°, the speed deviation decreases from 0.027% to -0.011%. This indicates an improvement in transmission smoothness; the output speed becomes more stable. Interestingly, when $\delta$ is further increased to 7°, the speed deviation increases slightly to -0.007%. This suggests that there exists an optimal range for the modification parameter $\delta_0$ (around 5° for this specific gear design). Within this range, the modification effectively improves stability, but beyond it, the benefits may diminish or the dynamic performance could degrade.
Influence of Modification Parameter on Dynamic Meshing Force
The dynamic meshing force between gear teeth is a direct source of vibration and noise in gear transmission systems. Analyzing its magnitude and variation is crucial for assessing the effectiveness of a modification design in reducing dynamic excitation. High or sharply fluctuating meshing forces lead to increased vibration and noise.
To investigate the impact of the cutter tilt angle $\delta$ on the dynamic meshing force of the VH-CATT cylindrical gear pair, simulations are conducted for a wider range of $\delta$ values: 0°, 1°, 2°, 3°, 4°, 5°, 6°, and 7°. The same loading condition (1500 N·m) is maintained. The contact force calculated by Adams between the two gear bodies is recorded as the dynamic meshing force. The time-history plots of the meshing force for each case are analyzed, and the maximum, minimum, and average meshing forces over a stable meshing cycle are extracted.
| Cutter Tilt Angle $\delta$ (°) | Max Force (N) | Min Force (N) | Avg Force (N) |
|---|---|---|---|
| 0 | 30866.0 | 3258.5 | 17062.3 |
| 1 | 30304.5 | 3471.8 | 16888.2 |
| 2 | 30550.8 | 2245.0 | 16397.9 |
| 3 | 30143.9 | 268.0 | 15606.0 |
| 4 | 29942.2 | 562.0 | 15252.1 |
| 5 | 29829.3 | 356.8 | 15093.1 |
| 6 | 29261.3 | 289.1 | 14775.2 |
| 7 | 30913.9 | 1860.6 | 16387.3 |
The data clearly shows a significant influence of the modification parameter on the meshing forces. As the cutter tilt angle $\delta$ increases from 0° up to 6°, the average dynamic meshing force shows a consistent decreasing trend, dropping from approximately 17062 N for the unmodified case ($\delta=0°$) to about 14775 N for $\delta=6°$. This represents a reduction of over 13% in the average force. The maximum force also shows a general decreasing trend, and the minimum force becomes very small, indicating smoother engagement with less impact. This reduction in average and peak forces is highly beneficial for reducing system vibration, stress, and wear.
However, when $\delta$ is increased to 7°, the average meshing force rises to about 16387 N, and the maximum force also increases significantly compared to the case with $\delta=6°$. This observation aligns with the trend seen in speed fluctuation analysis. It confirms that an optimal modification magnitude exists. Up to a certain value $\delta_0$ (which appears to be between 5° and 6° for this specific cylindrical gear design), the tilt modification effectively optimizes the contact pattern and load distribution across the tooth face, leading to lower and more stable meshing forces. Exceeding this optimal value alters the contact conditions unfavorably, potentially causing edge contact or other adverse effects, which in turn increases the dynamic forces.
Conclusion
This work presents a comprehensive analysis of the dynamic meshing performance of modified Variable Hyperbolic Circular-Arc-Tooth-Trace (VH-CATT) cylindrical gears. A tooth surface modification method based on tilting the large milling cutter head during the gear generation process is proposed. The mathematical model for the modified tooth surface of the VH-CATT cylindrical gear is rigorously derived based on gear meshing theory. Using this model, precise three-dimensional solid models of gear pairs with different cutter tilt angles ($\delta$) are developed. Multi-body dynamics simulations are then performed in the Adams software environment to analyze the dynamic behavior of these modified cylindrical gear pairs under load.
The simulation results yield clear insights into the effect of the modification parameter. The average output speed of the driven cylindrical gear remains largely unaffected by the modification, confirming the preservation of the intended speed ratio. However, the modification significantly influences the dynamic performance metrics. Both the speed fluctuation of the driven gear and the magnitude of the dynamic meshing force exhibit a non-monotonic relationship with the cutter tilt angle $\delta$. As $\delta$ increases from zero to an optimal value $\delta_0$ (identified in this study to be approximately in the range of 5° to 6° for the given cylindrical gear parameters), the amplitude of speed fluctuation decreases, and the average as well as peak dynamic meshing forces are reduced. This demonstrates the effectiveness of the proposed tilt modification in improving transmission smoothness and reducing dynamic excitations for the VH-CATT cylindrical gear.
However, further increasing $\delta$ beyond this optimal range leads to a degradation in performance, with increased speed fluctuation and higher meshing forces. This underscores the importance of selecting an appropriate modification magnitude. The research establishes a methodology for the design and dynamic analysis of modified VH-CATT cylindrical gears. The findings provide a valuable theoretical foundation for the vibration reduction, noise control, and performance optimization of this promising type of cylindrical gear transmission system through targeted tooth surface modification.